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Book review

Author(s) Bliedtner, J.
Hansen, W.
Title Potential Theory. An Analytic and Probabilistic Approach to Balayage
Publisher Springer-Verlag
Year of publication 1986
Reviewed by P.P. Zabrejko

There are lot of books devoted to one of remarkable modern theories in mathematics, namely, the potential theory. Amongst them one can find either elementary and introductory textbooks or serious and fundamental monographs. It is sufficient to bring to mind the books by O.D. Kellog, H. Cartan, W. Meier, J.L. Doob, G.A. Hunt, L.L. Helms, M. Brelot, C. Constantinescu - A. Cornea, R.M Blumenthal - R.K Getoor, J. Werer, M. Tsuji, M. Kac, C. Delacherie - P.A. Meyer, B.-W. Schulze - G. Wildenhain ... However and a new book on potential theory will be able to find its place in this row and turn out to be useful either young investigators or serious mathematicians in field.

There are two different approaches to modern potential theory. The analytic approaches is based on analysis of the classical Dirichlet problem for Laplace equation or general elliptic equations of second and high order. The probabilistic approach is based on analysis on the theory of stochastic processes. Both theories are linked with Laplace or more generally elliptic differential operator and one can find numerous analogs in both theories; so the harmonic functions and measures in Dirichlet problem correspond to hitting distributions for Brownian motion or in other level the positive hyperharmonic functions for elliptic equation are the excessive functions of corresponding Brownian semigroup. However both analytical and probabilistic theories developed almost independently. Moreover for specialists in partial differential equations it is hard to understand and estimate results by their colleagues in probabilistic potential theory and vice versa these colleagues have troubles with results by first ones.

The book by J. Bliedtner and W. Hansen is one of the first attempts to unit both theories on the base of the balayage theory that date to H. Poincare. More exactly the authors write "This book ... has two aims. The first is to give a comprehensive account of the close connection between analytic and probabilistic potential theory with the notion of a balayage space appearing as a natural link. The second aim is to demonstrate the fundamental importance of those concept by using it to give a straight presentation of balayage theory which in turn is then applied to the Dirichlet problem". The authors' approach have allowed them to embrace the main concepts of both theories including Riesz potentials, Markov chains and numerous others. The main part of the book could only be found in the original papers.

The book contains Introductory, 9 chapters, Notes, Bibliography, Index of Symbols, Subject Index and Guide to Standard Examples. Chapter 0 "Classical Potential Theory" presents the discussion of relations between positive hyperharmonic functions and excessive functions for the Brownian semigroup. The main goal of this introductory chapter to give motivation for further constructions in the book. Chapter 1 "General Preliminaries" gives some necessary prerequisites: function cones, Choquet boundaries analytic sets, capacitances and so on.

Chapters 2 "Excessive functions", 3 "Hyperharmonic Functions" and 4 "Markov Processes" are devoted to some aspects of the theory of Markov processes. Here the balayage space is appeared with some natural way and explicit constructions of kernels, resolvents and semigroups for subMarkov semigroups are given, harmonic structure of a balayage space and its different properties as convergence and sheaf ones, minimum principle are discussed, different types of Markov processes are described and so on. The main result here is the equivalence of four different descriptions of these processes: on the base of balayage space, on the base of families of harmonic kernels, on the base of subMarkov semigroups and on the base of Markov processes. All these considerations are illustrated on three standard examples, namely, classical potential theory, translation in R and discrete potential theory. Chapter 5 "Examples" deals with more complicated examples: Riesz potentials, heat equations, Brownian semigroups on the infinite dimensional torus and some others.

Chapter 6 "Balayage Theory" is central in the book and deals with balayage of functions and measures. The identification of balayage measures and hitting distributions of associated Hunt processes, the fine topology and finely continuous functions, various types of exceptional sets, the fine support of continuous potentials, the converge properties of balayage measures, the set of extreme representating measures are consequently discussed here. All proof are analytical however the probabilistic interpretations of problems considered play important role. Chapter 7 "Dirichlet Problem" deals with different types of Dirichlet problems. First the Perron-Wiener-Brelot method is adopted to balayage space and then a Choquet type theory is developed; at the end of chapter the weak Dirichlet problem, fine Dirichlet problem, some special approximation theorems and removable singularities are studied.

The final chapter 8 "Partial Differential Equations" deals with elliptic and parabolic differential equations of second type; it turns out that the general theory presented in the main part of the book covers the potential theory for such equations.

From the preceding one can see interesting conception of the book. To increase the familiarity of the reader with the material nearly all of the sections contain numerous exercises, bibliographical notes including some historical remarks, symbol and subject index, guide to standard examples are added for the sake of convenience. The reader is assumed to be familiar with basic facts from functional analysis, measure and probability theory, but no knowledge of potential theory is presupposed. Of course in places it is not easy to read the book however, undoubtedly, it must be reccomended to graduate students and experts for reading and studying. The first could be acquaintance with one of the beautiful theories of modern mathematics, the latter could get some pleasure from the new approach of authors to so habitual and investigated objects of mathematics.